Abstract
Deformations of commutative rings of self-adjoint ordinary differential operators of rank 2 given by soliton equations are studied.
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I. M. Krichever and S. P. Novikov, “Holomorphic bundles over algebraic curves and nonlinear equations,” UspekhiMat. Nauk 35 (6), 47–68 (1980) [RussianMath. Surveys 35 (6), 53–79 (1980)].
I. M. Krichever and S. P. Novikov, “Two-dimensionalized Toda lattice, commuting difference operators, and holomorphic vector bundles,” Uspekhi Mat. Nauk 58 (3), 51–88 (2003) [Russian Math. Surveys 58 (3), 473–510 (2003)].
V. G. Drinfeld and V. V. Sokolov, “Symmetries in the Lax equations,” in Integrable Systems, Ed. by A. B. Shabat (Ufa, 1982) [in Russian].
I. M. Krichever, “Commutative rings of ordinary linear differential operators,” Funktsional. Anal. Prilozhen. 12 (3), 20–31 (1978).
P. Dehornoy, “Operateurs differentiels et courbes elliptiques,” Compositio Math. 43 (1), 71–99 (1981).
P. G. Grinevich, “Rational solutions for the equation of commutation of differential operators,” Funktsional. Anal. Prilozhen. 16 (1), 19–24 (1982) [Functional Anal. Appl. 16 (1), 15–19 (1982)].
P. G. Grinevich and S. P. Novikov, “Spectral theory of commuting operators of rank two with periodic coefficients,” Funktsional. Anal. Prilozhen. 16 (1), 25–26 (1982) [Functional Anal. Appl. 16 (1), 19–20 (1982)].
F. A. Grünbaum, “Commuting pairs of linear ordinary differential operators of orders four and six,” Phis. D 31 (3), 424–433 (1988).
G. A. Latham, “Rank 2 commuting ordinary differential operators and Darboux conjugates of KdV,” Appl. Math. Lett 8 (6), 73–78 (1995).
G. Latham and E. Previato, “Darboux transformations for higher-rank Kadomtsev–Petviashvili and Krichever–Novikov equations,” Acta Appl.Math. 39 (1–3), 405–433 (1995).
E. Previato, “Seventy years of spectral curves: 1923–1993,” in Integrable Systems at Quantum Groups, Lecture Notes in Math. (Springer-Verlag, Berlin, 1996), Vol. 1620, pp. 419–481.
E. Previato and G. Wilson, “Differential operators and rank 2 bundles over elliptic curves,” Compositio Math. 81 (1), 107–119 (1992).
O. I. Mokhov, “Commuting differential operators of rank 3 and nonlinear differential equations,” Izv. Akad. Nauk SSSR Ser. Mat. 53 (6), 1291–1315 (1989) [Math. USSR-Izv. 35 (3), 629–655 (1990)].
V. N. Davletshina, “On self-adjoint commuting differential operators of rank two,” Sib. Élektron. Mat. Izv. 10, 109–112 (2013).
A. E. Mironov, “Self-adjoint commuting ordinary differential operators,” Invent. Math. 197 (2), 417–431 (2014).
A. E.Mironov, “Periodic and rapid decay rank 2 self-adjoint commuting differential operators,” Amer.Math. Soc. Transl. Ser. 2 234, 309–322 (2014).
A. E.Mironov, “On a ring of commutative differential operators of rank two, which corresponds to a curve of genus two,” Mat. Sb. 195 (5), 103–114 (2004) [Sb.Math. 195 (5), 711–722 (2004)].
A. E.Mironov, “Commuting rank 2 differential operators corresponding to a curve of genus 2,” Funktsional. Anal. Prilozhen. 39 (3), 91–94 (2005) [Functional Anal. Appl. 39 (3), 240–243 (2005)].
O. I.Mokhov, “On commutative subalgebras of theWeyl algebra related to commuting operators of arbitrary rank and genus,” Mat. Zametki 94 (2), 314–316 (2013) [Math. Notes 94 (1–2), 298–300 (2013)].
D. Zuo, “Commuting differential operators of rank 3 associated to a curve of genus 2,” SIGMA 8 (044) (2012).
B. A. Dubrovin, V. B. Matveev, and S. P. Novikov, “Non-linear equations of Korteweg–deVries type, finite-zone linear operators, and Abelian varieties,” Uspekhi Mat. Nauk 31 (1), 55–136 (1976) [Russian Math. Surveys 31 (1), 59–146 (1976)].
D. P. Novikov, “Algebraic-geometric solutions of the Krichever–Novikov equation,” Teoret. Mat. Fiz. 121 (3), 367–373 (1999) [Theoret. and Math. Phys. 121 (3), 1567–1573 (1999)].
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Davletshina, V.N. Self-adjoint commuting differential operators of rank 2 and their deformations given by soliton equations. Math Notes 97, 333–340 (2015). https://doi.org/10.1134/S0001434615030049
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DOI: https://doi.org/10.1134/S0001434615030049